List of top Mathematics Questions on Three Dimensional Geometry asked in VITEEE

Find the value of \(k\) if the lines \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x-4}{k} = \frac{y-1}{2} = \frac{z}{1} \) are perpendicular.

What is the angle between the two straight lines \[ y = (2 - \sqrt{3})x + 5 \quad \text{and} \quad y = (2 + \sqrt{3})x - 7? \]

If the angle \( \theta \) between the line \[ \frac{x + 1}{2} = \frac{z - 2}{2} = \frac{y + 4}{\sqrt{n}} \] and the plane \( 2x - y + z + 4 = 0 \) is such that \( \sin \theta = \frac{1}{3} \), then the value of \( n \) is:

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000 if the rate of growth of bacteria is proportional to the number present?

If \( (-4, 5) \) is one vertex and \( 7x - y + 8 = 0 \) is one diagonal of a square, then the equation of second diagonal is:

If a line segment OP makes angles of $ \frac{\pi}{4}$ and $\frac{\pi }{3}$ with X-axis and Y-axis, respectively. Then, the direction cosines are

The two lines $x = my + n, z = py + q$ and $x = m'y + n',\, z = p'y + q'$ are perpendicular to each other, if

If $(2, 7, 3)$ is one end of a diameter of the sphere $x^2 + y^2 + z^2 - 6\,x -12\,y -2\,z + 20 = 0$, then the coordinates of the other end of the diameter are

If the points \( (1, 2, 3) \) and \( (2, -1, 0) \) lie on the opposite sides of the plane \( 2x + 3y - 2z = k \), then \( k \) is:

If a plane meets the coordinate axes at \( A, B, C \) such that the centroid of the triangle is \( (1, 2, 4) \), then the equation of the plane is?

The volume of the tetrahedron included between the plane \( 3x + 4y - 5z = 60 \) and the coordinate planes is?

The vertices of a triangle are \( A(0,4,1) \), \( B(2,-3,-1) \), and \( C(4,5,0) \), then the orthocenter of ABC is?

If the points $(1, 2, 3)$ and $(2, -1, 0)$ lie on the opposite sides of the plane $2x + 3y - 2z = k,$ then

If a plane meets the coordinate axes at $A,B$ and $C$ such that the centroid of the triangle is $(1, 2, 4)$, then the equation of the plane is

Two lines $ \frac{x-1}{2} = \frac{y+1}{3} = \frac{z -1}{4}$ and $ \frac{x-3}{1} = \frac{y-k}{2} = z $ intersect at a point, if $k$ is equal to

The distance between the line \(\vec{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda(\hat{i}-\hat{j}+4\hat{k})\) and the plane \(\vec{r}\cdot(\hat{i}+5\hat{j}+\hat{k})=5\), is

The equation of sphere concentric with the sphere \(x^2+y^2+z^2-4x-6y-8z-5=0\) and which passes through the origin, is

If the lines \(\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-1}{4}\) and \(\dfrac{x-3}{1}=\dfrac{y-k}{2}=\dfrac{z}{1}\) intersect, then the value of \(k\) is

The perimeter of the triangle with vertices at \((1,0,0),(0,1,0)\) and \((0,0,1)\) is

If a line in the space makes angles \(\alpha,\beta,\gamma\) with the coordinate axes, then
\[ \cos2\alpha+\cos2\beta+\cos2\gamma+\sin^2\alpha+\sin^2\beta+\sin^2\gamma \] equals

The radius of the sphere
\[ x^2+y^2+z^2=12x+4y+3z \] is

The angle between the lines whose direction cosines satisfy the equations
\[ l+m+n=0,\quad l^2+m^2-n^2=0 \] is

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0, l^{2}+m^{2}-n^{2}=0$ is

The plane through the point \((-1,-1,-1)\) and containing the line of intersection of the planes \(\vec{r}\cdot(\hat{i}+3\hat{j}-\hat{k})=0\) and \(\vec{r}\cdot(\hat{i}+2\hat{k})=0\) is